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We analyze the coefficients of partition functions of Vafa-Witten theory forthe complex projective plane $\mathbb{CP}^2$. We experimentally study thegrowth of the coefficients for gauge group $SU(2)$ and $SU(3)$, which areexamples of mock modular forms of depth $1$ and 2 respectively. We alsointroduce the notion of ``mock cusp form'', and study an example of weight 3related to the $SU(3)$ partition function. Numerical experiments on the first200 coefficients suggest that the coefficients of a mock modular form of weight$k$ grow as the coefficients of a modular form of weight $k$, that is to say as$n^{k-1}$. On the other hand the coefficients of the mock cusp form appear togrow as $n^{3/2}$, which exceeds the growth of classical cusp forms of weight3. We provide bounds using saddle point analysis, which however largely exceedthe experimental observation.

Numerical experiments on coefficients of instanton partition functions